Newton's law of gravitation resembles Coulomb's law of electrical forces, which is used to calculate the magnitude of electrical force arising between two charged bodies. Both are inverse-square laws, where force is inversely proportional to the square of the distance between the bodies. Coulomb's law has the product of two charges in place of the product of the masses, and the electrostatic constant in place of the gravitational constant.

Newton's law has since been superseded by Einstein's theory of general relativity, but it continues to be used as an excellent approximation of the effects of gravity in most applications. Relativity is required only when there is a need for extreme precision, or when dealing with very strong gravitational fields, such as those found near extremely massive and dense objects, or at very close distances (such as Mercury's orbit around the sun).

History

Early history

A recent assessment (by Ofer Gal) about the early history of the inverse square law is "by the late 1660s", the assumption of an "inverse proportion between gravity and the square of distance was rather common and had been advanced by a number of different people for different reasons". The same author does credit Hooke with a significant and even seminal contribution, but he treats Hooke's claim of priority on the inverse square point as uninteresting since several individuals besides Newton and Hooke had at least suggested it, and he points instead to the idea of "compounding the celestial motions" and the conversion of Newton's thinking away from "centrifugal" and towards "centripetal" force as Hooke's significant contributions.

Plagiarism dispute

In 1686, when the first book of Newton's Principia was presented to the Royal Society, Robert Hooke accused Newton of plagiarism by claiming that he had taken from him the "notion" of "the rule of the decrease of Gravity, being reciprocally as the squares of the distances from the Center". At the same time (according to Edmond Halley's contemporary report) Hooke agreed that "the Demonstration of the Curves generated thereby" was wholly Newton's.[5]

In this way the question arose as to what, if anything, Newton owed to Hooke. This is a subject extensively discussed since that time and on which some points continue to excite some controversy.

Hooke's work and claims

Robert Hooke published his ideas about the "System of the World" in the 1660s, when he read to the Royal Society on 21 March 1666 a paper "On gravity", "concerning the inflection of a direct motion into a curve by a supervening attractive principle", and he published them again in somewhat developed form in 1674, as an addition to "An Attempt to Prove the Motion of the Earth from Observations".[6] Hooke announced in 1674 that he planned to "explain a System of the World differing in many particulars from any yet known", based on three "Suppositions": that "all Celestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers" [and] "they do also attract all the other Celestial Bodies that are within the sphere of their activity";[7] that "all bodies whatsoever that are put into a direct and simple motion, will so continue to move forward in a straight line, till they are by some other effectual powers deflected and bent..."; and that "these attractive powers are so much the more powerful in operating, by how much the nearer the body wrought upon is to their own Centers". Thus Hooke clearly postulated mutual attractions between the Sun and planets, in a way that increased with nearness to the attracting body, together with a principle of linear inertia.

Hooke's statements up to 1674 made no mention, however, that an inverse square law applies or might apply to these attractions. Hooke's gravitation was also not yet universal, though it approached universality more closely than previous hypotheses.[8] He also did not provide accompanying evidence or mathematical demonstration. On the latter two aspects, Hooke himself stated in 1674: "Now what these several degrees [of attraction] are I have not yet experimentally verified"; and as to his whole proposal: "This I only hint at present", "having my self many other things in hand which I would first compleat, and therefore cannot so well attend it" (i.e. "prosecuting this Inquiry").[6] It was later on, in writing on 6 January 1679|80[9] to Newton, that Hooke communicated his "supposition ... that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall, and Consequently that the Velocity will be in a subduplicate proportion to the Attraction and Consequently as Kepler Supposes Reciprocall to the Distance."[10] (The inference about the velocity was incorrect.[11])

Hooke's correspondence of 1679-1680 with Newton mentioned not only this inverse square supposition for the decline of attraction with increasing distance, but also, in Hooke's opening letter to Newton, of 24 November 1679, an approach of "compounding the celestial motions of the planets of a direct motion by the tangent & an attractive motion towards the central body".[12]

Newton's work and claims

Newton, faced in May 1686 with Hooke's claim on the inverse square law, denied that Hooke was to be credited as author of the idea. Among the reasons, Newton recalled that the idea had been discussed with Sir Christopher Wren previous to Hooke's 1679 letter.[13] Newton also pointed out and acknowledged prior work of others,[14] including Bullialdus,[15] (who suggested, but without demonstration, that there was an attractive force from the Sun in the inverse square proportion to the distance), and Borelli[16] (who suggested, also without demonstration, that there was a centrifugal tendency in counterbalance with a gravitational attraction towards the Sun so as to make the planets move in ellipses). D T Whiteside has described the contribution to Newton's thinking that came from Borelli's book, a copy of which was in Newton's library at his death.[17]

Newton further defended his work by saying that had he first heard of the inverse square proportion from Hooke, he would still have some rights to it in view of his demonstrations of its accuracy. Hooke, without evidence in favor of the supposition, could only guess that the inverse square law was approximately valid at great distances from the center. According to Newton, while the 'Principia' was still at pre-publication stage, there were so many a-priori reasons to doubt the accuracy of the inverse-square law (especially close to an attracting sphere) that "without my (Newton's) Demonstrations, to which Mr Hooke is yet a stranger, it cannot believed by a judicious Philosopher to be any where accurate."[18]

This remark refers among other things to Newton's finding, supported by mathematical demonstration, that if the inverse square law applies to tiny particles, then even a large spherically symmetrical mass also attracts masses external to its surface, even close up, exactly as if all its own mass were concentrated at its center. Thus Newton gave a justification, otherwise lacking, for applying the inverse square law to large spherical planetary masses as if they were tiny particles.[19] In addition, Newton had formulated in Propositions 43-45 of Book 1,[20] and associated sections of Book 3, a sensitive test of the accuracy of the inverse square law, in which he showed that only where the law of force is accurately as the inverse square of the distance will the directions of orientation of the planets' orbital ellipses stay constant as they are observed to do apart from small effects attributable to inter-planetary perturbations.

In regard to evidence that still survives of the earlier history, manuscripts written by Newton in the 1660s show that Newton himself had arrived by 1669 at proofs that in a circular case of planetary motion, "endeavour to recede" (what was later called centrifugal force) had an inverse-square relation with distance from the center.[21] After his 1679-1680 correspondence with Hooke, Newton adopted the language of inward or centripetal force. According to Newton scholar J. Bruce Brackenridge, although much has been made of the change in language and difference of point of view, as between centrifugal or centripetal forces, the actual computations and proofs remained the same either way. They also involved the combination of tangential and radial displacements, which Newton was making in the 1660s. The lesson offered by Hooke to Newton here, although significant, was one of perspective and did not change the analysis.[22] This background shows there was basis for Newton to deny deriving the inverse square law from Hooke.

Newton's acknowledgment

On the other hand, Newton did accept and acknowledge, in all editions of the 'Principia', that Hooke (but not exclusively Hooke) had separately appreciated the inverse square law in the solar system. Newton acknowledged Wren, Hooke and Halley in this connection in the Scholium to Proposition 4 in Book 1.[23] Newton also acknowledged to Halley that his correspondence with Hooke in 1679-80 had reawakened his dormant interest in astronomical matters, but that did not mean, according to Newton, that Hooke had told Newton anything new or original: "yet am I not beholden to him for any light into that business but only for the diversion he gave me from my other studies to think on these things & for his dogmaticalness in writing as if he had found the motion in the Ellipsis, which inclined me to try it ..."[14]

Modern controversy

Since the time of Newton and Hooke, scholarly discussion has also touched on the question of whether Hooke's 1679 mention of 'compounding the motions' provided Newton with something new and valuable, even though that was not a claim actually voiced by Hooke at the time. As described above, Newton's manuscripts of the 1660s do show him actually combining tangential motion with the effects of radially directed force or endeavour, for example in his derivation of the inverse square relation for the circular case. They also show Newton clearly expressing the concept of linear inertia—for which he was indebted to Descartes' work, published in 1644 (as Hooke probably was).[24] These matters do not appear to have been learned by Newton from Hooke.

Nevertheless, a number of authors have had more to say about what Newton gained from Hooke and some aspects remain controversial.[25] The fact that most of Hooke's private papers had been destroyed or have disappeared does not help to establish the truth.

Newton's role in relation to the inverse square law was not as it has sometimes been represented. He did not claim to think it up as a bare idea. What Newton did was to show how the inverse-square law of attraction had many necessary mathematical connections with observable features of the motions of bodies in the solar system; and that they were related in such a way that the observational evidence and the mathematical demonstrations, taken together, gave reason to believe that the inverse square law was not just approximately true but exactly true (to the accuracy achievable in Newton's time and for about two centuries afterwards – and with some loose ends of points that could not yet be certainly examined, where the implications of the theory had not yet been adequately identified or calculated).[26][27]

About thirty years after Newton's death in 1727, Alexis Clairaut, a mathematical astronomer eminent in his own right in the field of gravitational studies, wrote after reviewing what Hooke published, that "One must not think that this idea ... of Hooke diminishes Newton's glory"; and that "the example of Hooke" serves "to show what a distance there is between a truth that is glimpsed and a truth that is demonstrated".[28][29]

Assuming SI units, F is measured in newtons (N), m1 and m2 in kilograms (kg), r in meters (m), and the constant G is approximately equal to 6989667400000000000♠6.674×10−11 N m2 kg−2.[30] The value of the constant G was first accurately determined from the results of the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798, although Cavendish did not himself calculate a numerical value for G.[4] This experiment was also the first test of Newton's theory of gravitation between masses in the laboratory. It took place 111 years after the publication of Newton's Principia and 71 years after Newton's death, so none of Newton's calculations could use the value of G; instead he could only calculate a force relative to another force.

Bodies with spatial extent

Gravitational field strength within the Earth.

Gravity field near earth at 1,2 and A.

If the bodies in question have spatial extent (rather than being theoretical point masses), then the gravitational force between them is calculated by summing the contributions of the notional point masses which constitute the bodies. In the limit, as the component point masses become "infinitely small", this entails integrating the force (in vector form, see below) over the extents of the two bodies.

In this way it can be shown that an object with a spherically-symmetric distribution of mass exerts the same gravitational attraction on external bodies as if all the object's mass were concentrated at a point at its centre.[3] (This is not generally true for non-spherically-symmetrical bodies.)

For points inside a spherically-symmetric distribution of matter, Newton's Shell theorem can be used to find the gravitational force. The theorem tells us how different parts of the mass distribution affect the gravitational force measured at a point located a distance r0 from the center of the mass distribution:[31]

The portion of the mass that is located at radii r < r0 causes the same force at r0 as if all of the mass enclosed within a sphere of radius r0 was concentrated at the center of the mass distribution (as noted above).

The portion of the mass that is located at radii r > r0 exerts no net gravitational force at the distance r0 from the center. That is, the individual gravitational forces exerted by the elements of the sphere out there, on the point at r0, cancel each other out.

As a consequence, for example, within a shell of uniform thickness and density there is no net gravitational acceleration anywhere within the hollow sphere.

Furthermore, inside a uniform sphere the gravity increases linearly with the distance from the center; the increase due to the additional mass is 1.5 times the decrease due to the larger distance from the center. Thus, if a spherically symmetric body has a uniform core and a uniform mantle with a density that is less than 2/3 of that of the core, then the gravity initially decreases outwardly beyond the boundary, and if the sphere is large enough, further outward the gravity increases again, and eventually it exceeds the gravity at the core/mantle boundary. The gravity of the Earth may be highest at the core/mantle boundary.

Vector form

Field lines drawn for a point mass using 24 field lines

Gravity field surrounding Earth from a macroscopic perspective.

Gravity field lines representation is arbitrary as illustrated here represented in 30x30 grid to 0x0 grid and almost being parallel and pointing straight down to the center of the Earth

Gravity in a room: the curvature of the Earth is negligible at this scale, and the force lines can be approximated as being parallel and pointing straight down to the center of the Earth

Newton's law of universal gravitation can be written as a vectorequation to account for the direction of the gravitational force as well as its magnitude. In this formula, quantities in bold represent vectors.

It can be seen that the vector form of the equation is the same as the scalar form given earlier, except that F is now a vector quantity, and the right hand side is multiplied by the appropriate unit vector. Also, it can be seen that F12 = −F21.

Gravitational field

The gravitational field is a vector field that describes the gravitational force which would be applied on an object in any given point in space, per unit mass. It is actually equal to the gravitational acceleration at that point.
It is a generalization of the vector form, which becomes particularly useful if more than 2 objects are involved (such as a rocket between the Earth and the Moon). For 2 objects (e.g. object 2 is a rocket, object 1 the Earth), we simply write r instead of r12 and m instead of m2 and define the gravitational field g(r) as:

so that we can write:

This formulation is dependent on the objects causing the field. The field has units of acceleration; in SI, this is m/s2.

Gravitational fields are also conservative; that is, the work done by gravity from one position to another is path-independent. This has the consequence that there exists a gravitational potential field V(r) such that

If m1 is a point mass or the mass of a sphere with homogeneous mass distribution, the force field g(r) outside the sphere is isotropic, i.e., depends only on the distance r from the center of the sphere. In that case

the gravitational field is on, inside and outside of symmetric masses.

As per Gauss Law, field in a symmetric body can be found by the mathematical equation:

where is a closed surface and is the mass enclosed by the surface.

Hence, for a hollow sphere of radius and total mass ,

For a uniform solid sphere of radius and total mass ,

Problematic aspects

Newton's description of gravity is sufficiently accurate for many practical purposes and is therefore widely used. Deviations from it are small when the dimensionless quantities φ/c2 and (v/c)2 are both much less than one, where φ is the gravitational potential, v is the velocity of the objects being studied, and c is the speed of light.[32] For example, Newtonian gravity provides an accurate description of the Earth/Sun system, since

where rorbit is the radius of the Earth's orbit around the Sun.

In situations where either dimensionless parameter is large, then general relativity must be used to describe the system. General relativity reduces to Newtonian gravity in the limit of small potential and low velocities, so Newton's law of gravitation is often said to be the low-gravity limit of general relativity.

Theoretical concerns with Newton's expression

There is no immediate prospect of identifying the mediator of gravity. Attempts by physicists to identify the relationship between the gravitational force and other known fundamental forces are not yet resolved, although considerable headway has been made over the last 50 years (See: Theory of everything and Standard Model). Newton himself felt that the concept of an inexplicable action at a distance was unsatisfactory (see "Newton's reservations" below), but that there was nothing more that he could do at the time.

Newton's theory of gravitation requires that the gravitational force be transmitted instantaneously. Given the classical assumptions of the nature of space and time before the development of General Relativity, a significant propagation delay in gravity leads to unstable planetary and stellar orbits.

Observations conflicting with Newton's formula

Newton's Theory does not fully explain the precession of the perihelion of the orbits of the planets, especially of planetMercury, which was detected long after the life of Newton.[33] There is a 43 arcsecond per century discrepancy between the Newtonian calculation, which arises only from the gravitational attractions from the other planets, and the observed precession, made with advanced telescopes during the 19th Century.

The predicted angular deflection of light rays by gravity that is calculated by using Newton's Theory is only one-half of the deflection that is actually observed by astronomers. Calculations using General Relativity are in much closer agreement with the astronomical observations.

In spiral galaxies the orbiting of stars around their centers seems to strongly disobey to Newton's law of universal gravitation. Astrophysicists, however, explain this spectacular phenomenon in the framework of the Newton's laws, with the presence of large amounts of Dark matter.

The observed fact that the gravitational mass and the inertial mass is the same for all objects is unexplained within Newton's Theories. General Relativity takes this as a basic principle. See the Equivalence Principle. In point of fact, the experiments of Galileo Galilei, decades before Newton, established that objects that have the same air or fluid resistance are accelerated by the force of the Earth's gravity equally, regardless of their different inertial masses. Yet, the forces and energies that are required to accelerate various masses is completely dependent upon their different inertial masses, as can be seen from Newton's Second Law of Motion, F = ma.

Newton's reservations

While Newton was able to formulate his law of gravity in his monumental work, he was deeply uncomfortable with the notion of "action at a distance" which his equations implied. In 1692, in his third letter to Bentley, he wrote: "That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it."

He never, in his words, "assigned the cause of this power". In all other cases, he used the phenomenon of motion to explain the origin of various forces acting on bodies, but in the case of gravity, he was unable to experimentally identify the motion that produces the force of gravity (although he invented two mechanical hypotheses in 1675 and 1717). Moreover, he refused to even offer a hypothesis as to the cause of this force on grounds that to do so was contrary to sound science. He lamented that "philosophers have hitherto attempted the search of nature in vain" for the source of the gravitational force, as he was convinced "by many reasons" that there were "causes hitherto unknown" that were fundamental to all the "phenomena of nature". These fundamental phenomena are still under investigation and, though hypotheses abound, the definitive answer has yet to be found. And in Newton's 1713 General Scholium in the second edition of Principia: "I have not yet been able to discover the cause of these properties of gravity from phenomena and I feign no hypotheses... It is enough that gravity does really exist and acts according to the laws I have explained, and that it abundantly serves to account for all the motions of celestial bodies."[34]

Einstein's solution

These objections were explained by Einstein's theory of general relativity, in which gravitation is an attribute of curved spacetime instead of being due to a force propagated between bodies. In Einstein's theory, energy and momentum distort spacetime in their vicinity, and other particles move in trajectories determined by the geometry of spacetime. This allowed a description of the motions of light and mass that was consistent with all available observations. In general relativity, the gravitational force is a fictitious force due to the curvature of spacetime, because the gravitational acceleration of a body in free fall is due to its world line being a geodesic of spacetime.

Extensions

Newton was the first to consider in his Principia an extended expression of his law of gravity including an inverse-cube term of the form

, B a constant

attempting to explain the Moon's apsidal motion. Other extensions were proposed by Laplace (around 1790) and Decombes (1913):[35]

(Laplace)

(Decombes)

In recent years quests for non-inverse square terms in the law of gravity have been carried out by neutron interferometry.[36]

Solutions of Newton's law of universal gravitation

The n-body problem is an ancient, classical problem[37] of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem — from the time of the Greeks and on — has been motivated by the desire to understand the motions of the Sun, planets and the visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem too.[38] The n-body problem in general relativity is considerably more difficult to solve.
The classical physical problem can be informally stated as: given the quasi-steady orbital properties (instantaneous position, velocity and time)[39]of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times.[40]

Friday, September 4, 2015

CO2

For some years, climate activists have claimed that we can't wait until we have all the data on AGW/CC, because it will be too late by then. But it is 2015 now, and we do have substantial data on many of the trends climate models have been used to predict up until now. I have been working with some of this data from the NOAA, EPA, and 2015 satellite datasets, on global temperatures, methane, and CO2 levels. Not surprisingly, there are some excellent fits to trend lines, which can be extrapolated to 2100.A word of caution here, though: trend lines cannot predict events that may change them, and some lines simply don't make scientific sense. For example,

PPM CO2 Increases from 1959 - 2014

Clearly CO2 increases are not going to follow this 6'th order polynomial trend, although it did give the best fit of all trends I tried. This is understandable. CO2 increases vary considerably from year to year. Better is to follow atmospheric CO2 levels, which I have done below:

PPM CO2 Levels 1980

Notice I've used a quadratic trend line here, which fits with > 99.99% correlation. According to CO2 forcing calculations, this leads to an approximate 1.7 degree C increase in temperature above 2014, or 2.7 degrees overall since 1900. This fits well with official IPCC numbers, albeit on the low end. It also agrees with temperatures plotted from NOAA data:

The increase from 2014 to 2100 is about 1.6 degrees, or 2.6 degrees by the IPCC. Very encouraging!But not very hopeful, if many climate scientists are right. It's generally predicted that an additional rise above one degree from current or more may lead to unacceptable consequences, most of which I'm sure you've read about. Can we reduce them to acceptable levels? And in a way that doesn't crash the global economy -- no, civilization itself -- leading to billions dead and the survivors back in the Dark Ages? I think there is, and have tried to apply my thoughts to the above chart. The result is the new chart below:

I have used an exponential type decay on the CO2 increase model to project how emissions trends might be reduced down to virtually zero without (I hope) harming the global economy. This new trend would not only depend on political/economic conditions, but mainly on developments in science and technology..At present the trend is about 2.3 PPM/year, and will increase to 3.2 PPM by the end of this century if nothing is done. I calculate that this will reduce a projected CO2 increase from 240 PPM down to an additional mere 72 PPM. The calculated temperature increase from 2015 is then only about +0.5 degrees above 2014 conditions.

I must add, however, that even reducing carbon emissions to zero -- even removing the gas from the atmosphere -- will not mean atmospheric levels will drop quickly. About ten times as much CO2 is dissolved in the oceans than is in the air (the greatest majority is in living matter and carbonate rocks); so that, even if can reduce atmospheric levels, an equilibrium process will redistribute some of the dissolved gas back into the air. For example, to remove a billions tons of CO2 quickly and permanently from the latter might require removing almost 10 billion tons from the ocean in order to maintain equilibrium.

On the other hand, it is also likely that anthropogenic CO2 has not fully equilibrated with the oceans (it is supposed to have a half-life of ~100 years in the air), so that even if we added no more, it would probably absorb further, and atmospheric levels would slowly decline. I do not know how this might affect the 2100 CO2 levels should the scheme above be applied. http://www.nature.com/nature/journal/v488/n7409/full/nature11299.html

Methane

Of course, CO2is not the only greenhouse gas climate scientists have been worrying about. The other main other culprit has been methane. Therefor, I have plotted atmospheric methane levels from 1980-2014. Instead of plotting the gas directly, I have converted to CO2 PPM equivalents.

Satellite Data

The differences are striking. The satellite data indicate a full degree less warming by 2100 than the NOAA data, or only +0.7 degrees above current temperatures; however, the varriation is also much larger, so this might not be accurate. Incidentally, for those who follow this issue, the so-called "hiatus" (of "climate-denier" fame) is based on the satellite data (plotted from 1996-2014 below):

So the pause is real, at least for the satellite data. The fallacy, I think, is to assume that because temperatures have been falling for some eighteen years, we can declare global warming to be over. The entire data set does not support that, and it is common for global temperatures to decrease (and increase) over short periods. So beware.

I have puzzled over why the NOAA and satellite datasets differ as much as they do. Since NASA has both geostationary and polar satellites, it shouldn't be a lack of global coverage, though there could be systematic errors. There is also the often said accusation that the NOAA data have been "adjusted" (meaning fudged in this case) to make global warming appear worse than it is (and I have read about some peculiar adjustments), but I don't have no data to back this claim up. It could also be, of course, that satellites measure a higher region in the troposphere, while the NOAA data is strictly ground level.

Naturally, it could be any combination of any of these reasons.

Sea Level Rise and Global Ice

Now we have three conflicting data sets! CISRO (30 cm -> 2100) and NASA (40 cm) agree the best, and they are both best within predicted range for the middle of the next century (~ one meter rise from 1900). Of course, if CO2 emissions are reduced in the fashion described above, these numbers should not as high; I have no data to demonstrate this, however.

These data show an approximately 10,000 cubic kilometer loss in Arctic
ice over the last 35 years, leading to a three cm sea level rise during
that period. Unfortunately, I could not obtain the raw data for this
chart, so I cannot draw a trend line through the data (the straight line
came with the chart). We can, however, calculate a geometric increase
from this data, by assuming rising temperatures will double each
succeeding 35 year period. This give 2015-2050 = 20,000 km^3, 2050-2085
- 40,000 km^3, and 2085-2120 = 80,000 km^3. As we are only
extrapolating to 2100, this approximately adds an another 20,000 +
40,000 + 40,000 = 100,000 km^3 melted ice, enough to raise sea levels
another 30 cm over 2015. It is in excellent agreement with the CISPRO and NASA sea
level trendline above.

I haven't included Antarctic sea ice because, up to the present, it doesn't show enough shrinkage or expansion (estimates differ) to extrapolate a significant trend.

Very little, if any global sea ice area is demonstrated; however, if melting follows the path I have outlined above, it is quite possible that these levels will at least gradually shrink, raising global warming by reducing the planet's overall albedo.

Population Issues

It is, I hope, evident that the planet's increasing human population only adds more greenhouse gases, thereby increasing global warming. Of equal interest, however, is the per capital emission rates. To calculate this, I took the data from chart 2, and divided the points from the chart with population to produce the blue line:

The orange line is similar to that used to show how CO2 can be reduced to near zero before the century is out. See my comments about this modified trend above.

Conclusions

Global warming, along with the dire consequences this study supports, is very real and no hoax or scheme by some secret cabal of government and the scientific community. Yet neither is it an unstoppable catastrophe as some have alleged -- in fact, the trends revealed here mostly confirm to the more conservative estimates (model outputs) of how serious the problem is. That is no reason for complacency, however, for even the these models inexorably lead to serious problems.

I suggest, however, that these problems can be at least strongly ameliorated at least, if we invest in the basic science and technologies that alone can address the issue. For myself, I am strongly convinced that we will do so; more than that, that we have already begun.

About Me

My formal training is in chemistry. I also read a great deal of physics and biology. In fact I very much enjoy reading in general, mostly science, but also some fiction and history. I also enjoy computer programming and writing. I like hiking and exploring nature. I also enjoy people; not too much in social settings, but one on one; also, people with interesting or "off-beat" minds draw me to them. I also have some interest in Buddhism.

These days I get a lot more information from the internet, primarily through Wiki. Some television, e. g., documentaries, PBS shows like "Nova" and "Nature".

My favorite science writers are Jacob Bronowski ("The Ascent of Man") and Richard Dawkins (his "The Blind Watchmaker" is right up there up Ascent). I also have a favorite writer on Buddhism, Pema Chodron. Favorite films are "Annie Hall" (by Woody Allen), "The Maltese Falcon", "One Flew Over The Cuckoo's Nest", "As Good As It Gets", "Conspiracy Theory", Monty Python's "Search For The Holy Grail" and "Life of Brian", and a few others which I can't think about at the moment.

I love a number of classical works (Beethoven's "Pastoral", "Afternoon Of A Fawn" and "Clair De Lune" by Debussey , Pachelbel's "Canon" come to mind. My favorite piece is probably Gershwin's "Rhapsody in Blue". But I also enjoy a great deal in modern music, including many jazz pieces, folk songs by people like Dylan, Simon and Garfunkel, a hodgepodge of pieces by Crosby, Stills, and Nash, Niel Young, and practically everything the Beatles wrote.

My life over the last few years has been in some disarray, but I am finally "getting it together.". As I am very much into the sciences and writing, I would like to move more in this direction. I also enjoy teaching. As for my political leanings, most people would probably describe as basically liberal, though not extremely so. My religious leanings are to the absolutely none: I've alluded to my interest in Buddhism, but again this is not any supernatural or scientifically untested aspect of it but in the way it provides a powerful philosophy and set of practical, day to day methods of dealing with myself and the other human beings.